A generalized expression for the resistivity distribution in films: from the Young model to constant-phase element (CPE) behavior

The derivation of mathematical models accounting for the variation of physical properties along the thickness of thin systems such as oxide films, organic coatings and biological tissues, has been a challenging problem addressed for decades. Most attention has been paid to electrical resistivity because it typically undergoes a variation over several orders of magnitude, which is a necessary premise to explain the frequency dispersion over several decades of frequency observed in the impedance response of films. In this work we present a new distribution function for the spatial variation of resistivity within films, which is derived from the probability distribution that gives rise to exact constant-phase element (CPE) behavior. We demonstrate that this resistivity profile encompasses a wide variety of impedance behaviors including the probably most addressed types of frequency dispersion in films: the Young model and the CPE, thereby revealing an unknown relationship between them. These properties point to that distribution function as a promising fitting model for impedance data interpretation. In addition, the conditions for its applicability to real systems allowed us to postulate a lower bound for the CPE exponent as function of film properties which seems to be consistent with the values reported in the literature.